Theorem aaanv 44364

Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2331. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
aaanv	⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem aaanv

Step	Hyp	Ref	Expression
1		nfv 1913	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfv 1913	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	aaan 2331	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
4	3	bicomi 224	1 ⊢ ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥∀𝑦(𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-11 2156  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-nf 1783

This theorem is referenced by: (None)